Internet Accounts keeps asking for Google password

Question

I've enabled 2-step verification on my Google account.
I've had this account set up in Mail.app so it has an entry in my System Preferences > Internet Accounts
Now, with 2-step from Google, there are Application specific passwords. Mail.app can no longer use my Google account password. I had to log in to Google and create a password just for Mac Mail.app.
That worked just fine. Mail is now checking my Gmail for me.
When I open System Preferences > Internet Accounts and look at my Google account, it asks for my password. My Google ID password no longer works, as I'd expect. So I entered my Application Specific password for Mail.app.
This is the problem. It asks me three times, every time I want to look at my Google account details in System Preferences.
Sound familiar to anyone? Anyone solve this?
I tried creating an app specific password for System Preferences, but that didn't work.

Answer 1

I just had exactly the same issue!
Some extra info:
Happened only in 2 of my gmail account (have several)
One of them had 2nd factor auth enabled (account 1)
The other one use simply a password (account 2)
Both accounts accept the password on the 3rd try
for the account ,1 it is the application specific password
for the account, 2 it is the normal account password
Moreover:
account 1 uses: Mail, Contacts and Calendar
account 2 uses only Mail
Right now the issue in both accounts is 100% reproducible
Account 2's password is rather long (>18 chars) and complex (Upper, Lower, Number and Symbols)
This could indicate and issue, but I bet there many more user who had such extremes
Even if that was the issue, account 1's application specific password is pretty simple and of a fixed size
Seems like a bug on Apple's side...